Torath Kohanim / classical understanding regularly cited alongside TT’s material on the terumas hadeshen (change of garments).
ופשט את בגדיו ולבש בגדים אחרים — אינו חובה אלא דרך ארץ, שלא ילכלך בבגדיו שבא בהם תמיד את בגדיו שהוציא בהם את הדשן.
Ufashat es begadav velavash begadim acherim — eino chovah ela derech eretz, shelo yelachlech bebegadav sheba bahem tamid es begadav shehotzi bahem es hadeshen.
“And he shall take off his garments and put on other garments” — this is not an absolute obligation but a matter of proper conduct, so that he not soil with the ashes the garments in which he regularly serves.
Custom — procedural dignity (derech eretz).
When removing the ashes outside the camp the Kohen changes into other garments so as not to soil the garments of regular service.